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Question Number 107481 by ZiYangLee last updated on 11/Aug/20
Ifn∈Z+,showthat∑nk=1ln2(1+1k)<1
Answered by 1549442205PVT last updated on 11/Aug/20
∑nk=1ln2(1+1k)<1Wehaveln(1+x)<x∀x>0.Indeed,Considerthefunctionf(x)=x−ln(1+x)f′(x)=1−11+x=x1+x>0∀x>0⇒f(x)isincreaseon(0;+∞)Hencef(x)⩾f(0)=0∀x∈[0;+∞)⇔x>ln(1+x)∀x(0;+∞)(q.e.d)Hence,ln(1+1k)<1k⇒ln2(1+1k)<1k21(k+1)2<1k(k+1)=1k−1k+1.Therefore∑nk=1ln2(1+1k)=ln2(2)+ln2(32)+ln2(43)+ln2(54)+ln2(65)...+ln2(1+110)+1112+1122+...+1n2<ln2(2)+ln2(32)+ln2(43)+ln2(54)+ln2(65)...+ln2(1+110)+110−111+111−112+...+1n−1−1n=0.886300+110−1n<0.986300<1(q.e.d)
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